This course, "The Genetic Code of Mathematics – Decoding the Fundamental Elements," offers an in-depth and fascinating journey into the core of modern mathematics. Throughout the course, we will explore the foundational elements of mathematics, from the most basic concepts to sophisticated abstract structures.

We begin with a meticulous examination of addition and multiplication operations, while defining and thoroughly studying the fundamental number sets – natural, integer, rational, and irrational numbers. We will deepen our understanding of the relationships between these sets and learn to master unique mathematical notations, which constitute the formal language of mathematics.

Progressing to the formulation of properties and axioms – the basic assumptions of mathematics – we will examine which number sets satisfy these properties, leading us to the precise definition of the concept of a "field." We will demonstrate various fields, including the intriguing "modulo" field, and rigorously prove how they fulfill all field axioms.

Subsequently, we will delve into understanding the uniqueness of the real number field, defining the order axiom and demonstrating its importance. We will explore the concept of bounds – upper, lower, supremum, and infimum – and use them to define the completeness axiom, which distinguishes real numbers from other number sets.

Throughout the course, we will develop advanced mathematical thinking skills, including formal proof abilities, abstraction, and generalization. We will conclude with an overview of integral domains and their applications in advanced mathematics, opening a window to current research areas in pure and applied mathematics.

Understanding the fundamental elements of mathematics is critical for anyone engaged in advanced mathematics, theoretical physics, or computer science. This course provides the theoretical foundation necessary for comprehending more complex mathematical structures and serves as a first step towards abstract and precise mathematical thinking.

* The course material will be tailored to the age of the learners.